That's a relatively long focal length (not surprising). I would try...

1. Shoot at a speed of 1.5x your focal length or so. (so 300mm = 1/450th of a second)
2. Try not to zoom out to the absolute extreme of the lens... I know it's tempting to do so because it makes the moon take up more of the frame, but lenses will generally suffer some quality problems at their extremes, and from what I have seen many lenses seem to be sharpest at around 1/3 their max focal length.
3. Manual focus if possible... might be hard to do with the moon, but worth a shot.
4. Use a tripod if you can.

Sorry, gotta say this: Please do NOT follow the aperture and shutter speed advice that you have read on this thread. It is not good advice. Very very briefly, because I have addressed this before:

(1) Aperture should be as wide open as possible (lowest f/#) or POSSIBLY stopped down 2-3 stops because that's where many lenses are sharpest. Shooting at f/11, f/16, or other high numbers is silly and serves no purpose.

(3) Shutter speed should be fast but not too fast. In other words, fast enough to not overblow it but slow enough so that it's not too dark. Usually I have to make this point because people either up the ISO too much and so have a ridiculously fast speed, or they use too small an aperture so have to use a ridiculously slow speed.